This theorem and proof comes from Hoffman & Kunze (2ed) section 2.5 theorem 11.
I do not understand how the information preceding "It is now clear that R is uniquely determined by W" makes this theorem clear. Because I cannot quite pin point what I am missing, I will explain my understanding of the theorem up to the point where I cannot.
I think I understand that we start with a matrix whose row space is W. Then, we take the row-reduced echelon matrix. Then, we consider any (an arbitrary) row-reduced echelon matrix whose row space is W. Then, we know that the RREM row vectors form a basis for W. Thus, an arbitrary vector can be written as a linear combination of the basis vectors. After this point, however, I am lost as to how the information contributes to proving the theorem.
Any help would be greatly appreciated!
Best Answer
The proof.
I think you got the existence right:
As for the uniqueness, I will reformulate the proof slightly, hopefully, it will be easier to follow.
The rest relies on the properties of the RREF $R$: the position of the first non-zero entry of row $\rho_i$ is $k_i$, and its value is $1$. I shall call $\{k_i\}$ the basic positions.
As you said, $\beta\in W$ can be written as a linear combination of $\{\rho_i\}$. There are $r$ coefficients, which determine $\beta$. Because $\{\rho_i\}$ is such a special basis, these coefficients equal $b_{k_i}$. In particular, if an arbitrarily chosen $\beta \in W$ happens to satisfy $b_{k_s}=1$ and $b_{k_i}=0$ for $i\neq s$, then $\beta = \rho_s$.
This begs the following questions.
Can we recover the set $\{\rho_i\}$ by "fishing" the row-space $W$ for vectors that satisfy $b_{k_s}=1$ and $b_{k_i}=0,\,\forall i\neq s$?
Even if we "catch" $\{\rho_i\}$ and nothing more, does this prove the uniqueness of RREF?
The following observation removes the remaining obstacles.
If $\beta\in W$, $\beta\neq0$, then its first non-zero entry must be at a basic position (otherwise we contradict the expansion of $\beta$ in $\{\rho_i\}$). Conversely, for all basic positions, there exists a $\beta\in W$ with its first non-zero entry at this position (e.g., take $\beta=\rho_s$). In other words, the set of possible positions of the first non-zero entry of $\beta\in W$ is precisely the set of basic positions defined by the given RREM. This means that the set of basic positions $\{k_i\}$ is a property of $W$ and is the same for all RREMs.
We are certain we will "catch" $\{\rho_i\}$, nothing but $\{\rho_i\}$, and there is no ambiguity in defining $\{k_i\}$.
On a side note:
Because it does not!
The key thing to understand here is how the text is structured and how it should be read:
In general, the boundaries between these two types of paragraphs can be blurred, but in this text they are quite clear. Here, the only level-0 paragraph is
All other paragraphs are level-1:
So to answer your question, the information preceding "It is now clear ..." does not make it clear. You should look at the sentences that follow. I hope this helps!